Fundamentals of Hierarchical Linear and Multilevel Modeling 1 G
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Fundamentals of Hierarchical Linear and Multilevel Modeling 1 G. David Garson INTRODUCTION Hierarchical linear models and multilevel models are variant terms for what are broadly called linear mixed models (LMM). These models handle data where observations are not independent, correctly modeling correlated error. Uncorrelated error is an important but often violated assumption of statistical procedures in the general linear model family, which includes analysis of variance, correlation, regression, and factor analysis. Violations occur when error terms are not independent but instead cluster by one or more grouping variables. For instance, predicted student test scores and errors in predicting them may cluster by classroom, school, and municipality. When clustering occurs due to a grouping factor (this is the rule, not the exception), then the standard errors computed for prediction parameters will be wrong (ex., wrong b coefficients in regression). Moreover, as is shown in the application in Chapter 6 in this volume, effects of predictor variables may be misinterpreted, not only in magnitude but even in direction. Linear mixed modeling, including hierarchical linear modeling, can lead to substantially different conclusions compared to conventional regression analysis. Raudenbush and Bryk (2002), citing their 1988 research on the increase over time of math scores among students in Grades 1 through 3, wrote that with hierarchical linear modeling, The results were startling—83% of the variance in growth rates was between schools. In contrast, only about 14% of the variance in initial status was between schools, which is consistent with results typically encountered in cross-sectional studies of school effects. This analysis identified substantial differences among schools that conventional models would not have detected because such analyses do not allow for the partitioning of learning-rate variance into within- and between-school components. (pp. 9–10) 3 4 PART I. GUIDE Linear mixed models are a generalization of general linear models to better support analysis of a continuous dependent variable for the following: 1. Random effects: For when the set of values of a categorical predictor variable are seen not as the complete set but rather as a random sample of all values (ex., when the variable “product” has values representing only 30 of a possible 142 brands). Random effects modeling allows the researcher to make inferences over a wider population than is possible with regression or other general linear model (GLM) methods. 2. Hierarchical effects: For when predictor variables are measured at more than one level (ex., reading achievement scores at the student level and teacher–student ratios at the school level; or sentencing lengths at the offender level, gender of judges at the court level, and budgets of judicial districts at the district level). The researcher can assess the effects of higher levels on the intercepts and coefficients at the lowest level (ex., assess judge-level effects on predictions of sentencing length at the offender level). 3. Repeated measures: For when observations are correlated rather than independent (ex., before–after studies, time series data, matched-pairs designs). In repeated measures, the lowest level is the observation level (ex., student test scores on multiple occasions), grouped by observation unit (ex., students) such that each unit (student) has multiple data rows, one for each observation occasion. The versatility of linear mixed modeling has led to a variety of terms for the models it makes possible. Different disciplines favor one or another label, and different research targets influence the selection of terminology as well. These terms, many of which are discussed later in this chapter, include random intercept modeling, random coefficients modeling, random coefficients regression, random effects modeling, hierarchical linear modeling, multilevel modeling, linear mixed modeling, growth modeling, and longitudinal modeling. Linear mixed models in some disciplines are called “random effects” or “mixed effects” models. In economics, the term “random coefficient regression models” is used. In sociology, “multilevel modeling” is common, alluding to the fact that regression intercepts and slopes at the individual level may be treated as random effects of a higher (ex., organizational) level. And in statistics, the term “covariance components models” is often used, alluding to the fact that in linear mixed models one may decompose the covariance into components attributable to within-groups versus between-groups effects. In spite of many different labels, the commonality is that all adjust observation-level predictions based on the clustering of measures at some higher level or by some grouping variable. The “linear” in linear mixed modeling has a meaning similar to that in regres- sion: There is an assumption that the predictor terms on the right-hand side of the estimation equation are linearly related to the target term on the left-hand side. Of course, nonlinear terms such as power or log functions may be added to the predic- tor side (ex., time and time-squared in longitudinal studies). Also, the target variable may be transformed in a nonlinear way (ex., logit link functions). Linear mixed model (LMM) procedures that do the latter are “generalized” linear mixed models. CHAPTER 1. FUNDAMENtalS OF HIERARCHICAL LINEAR AND Multilevel MODELING 5 Just as regression and GLM procedures can be extended to “generalized general linear models” (GZLM), multilevel and other LMM procedures can be extended to “generalized linear mixed models” (GLMM), discussed further below. Linear mixed models for multilevel analysis address hierarchical data, such as when employee data are at level 1, agency data are at level 2, and department data are at level 3. Hierarchical data usually call for LMM implementation. While most multilevel modeling is univariate (one dependent variable), multivariate multilevel modeling for two or more dependent variables is available also. Likewise, models for cross-classified data exist for data that are not strictly hierarchical (ex., as when schools are a lower level and neighborhoods are a higher level, but schools may serve more than one neighborhood). The researcher undertaking causal modeling using linear mixed modeling should be guided by multilevel theory. That is, hierarchical linear modeling pos- tulates that there are cross-level causal effects. Just as regression models postulate direct effects of independent variables at level 1 on the dependent variable at level 1, so too, multilevel models specify cross-level interaction effects between vari- ables located at different levels. In doing multilevel modeling, the researcher postulates the existence of mediating mechanisms that cause variables at one level to influence variables at another level (ex., school-level funding may posi- tively affect individual-level student performance by way of recruiting superior teachers, made possible by superior financial incentives). Multilevel modeling tests multilevel theories statistically, simultaneously modeling variables at different levels without necessary recourse to aggrega- tion or disaggregation.1 Aggregation and disaggregation as used in regression models run the risk of ecological fallacy: What is true at one level need not be true at another level. For instance, aggregated state-level data on race and lit- eracy greatly overestimate the correlation of African American ethnicity with illiteracy because states with many African Americans tend to have higher illiteracy for all races. Individual-level data shows a low correlation of race and illiteracy. WHY USE LINEAR MIXED/HIERARCHICAL LINEAR/ MULTILEVEL MODELING? Why not just stick with tried-and-true regression models for analysis of data? The central reason, noted above, is that linear mixed models handle random effects, including the effects of grouping of observations under higher entities (ex., grouping of employees by agency, students by school, etc.). Clustering of observations within groups leads to correlated error terms, biased estimates of parameter (ex., regression coefficient) standard errors, and possible substantive mistakes when interpreting the importance of one or another predictor variable. Whenever data are sampled, the sampling unit as a grouping variable may well be a random effect. In a study of the federal bureaucracy, for instance, “agency” 6 PART I. GUIDE might be the sampling unit and error terms may cluster by agency, violating ordinary least squares (OLS) assumptions. Unlike OLS regression, linear mixed models take into account the fact that over many samples, different b coefficients for effects may be computed, one for each group. Conceptually, mixed models treat b coefficients as random effects drawn from a normal distribution of possible b’s, whereas OLS regression treats the b parameters as if they were fixed constants (albeit within a confidence inter- val). Treating “agency” as a random rather than fixed factor will alter and make more accurate the ensuing parameter estimates. Put another way, the misestima- tion of standard errors in OLS regression inflates Type 1 error (thinking there is relationship when there is not: false positives), whereas mixed models handle this potential problem. In addition, LMM can handle a random sampling vari- able like “agencies,” even when there are too many agencies to make into dummy variables in OLS regression and still expect reliable coefficients.